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Mirrors > Home > MPE Home > Th. List > cnncvsmulassdemo | Structured version Visualization version GIF version |
Description: Derive the associative law for complex number multiplication mulass 10347 interpreted as scalar multiplication to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
cnncvsmulassdemo | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . . . 4 ⊢ (ringLMod‘ℂfld) = (ringLMod‘ℂfld) | |
2 | 1 | cncvs 23321 | . . 3 ⊢ (ringLMod‘ℂfld) ∈ ℂVec |
3 | id 22 | . . . 4 ⊢ ((ringLMod‘ℂfld) ∈ ℂVec → (ringLMod‘ℂfld) ∈ ℂVec) | |
4 | 3 | cvsclm 23302 | . . 3 ⊢ ((ringLMod‘ℂfld) ∈ ℂVec → (ringLMod‘ℂfld) ∈ ℂMod) |
5 | 2, 4 | ax-mp 5 | . 2 ⊢ (ringLMod‘ℂfld) ∈ ℂMod |
6 | 1 | cnrbas 23318 | . . . 4 ⊢ (Base‘(ringLMod‘ℂfld)) = ℂ |
7 | 6 | eqcomi 2834 | . . 3 ⊢ ℂ = (Base‘(ringLMod‘ℂfld)) |
8 | cnfldex 20116 | . . . 4 ⊢ ℂfld ∈ V | |
9 | rlmsca 19568 | . . . 4 ⊢ (ℂfld ∈ V → ℂfld = (Scalar‘(ringLMod‘ℂfld))) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ ℂfld = (Scalar‘(ringLMod‘ℂfld)) |
11 | cnfldmul 20119 | . . . 4 ⊢ · = (.r‘ℂfld) | |
12 | rlmvsca 19570 | . . . 4 ⊢ (.r‘ℂfld) = ( ·𝑠 ‘(ringLMod‘ℂfld)) | |
13 | 11, 12 | eqtri 2849 | . . 3 ⊢ · = ( ·𝑠 ‘(ringLMod‘ℂfld)) |
14 | cnfldbas 20117 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
15 | 14 | eqcomi 2834 | . . . 4 ⊢ (Base‘ℂfld) = ℂ |
16 | 15 | eqcomi 2834 | . . 3 ⊢ ℂ = (Base‘ℂfld) |
17 | 7, 10, 13, 16 | clmvsass 23265 | . 2 ⊢ (((ringLMod‘ℂfld) ∈ ℂMod ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
18 | 5, 17 | mpan 681 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ‘cfv 6127 (class class class)co 6910 ℂcc 10257 · cmul 10264 Basecbs 16229 .rcmulr 16313 Scalarcsca 16315 ·𝑠 cvsca 16316 ringLModcrglmod 19537 ℂfldccnfld 20113 ℂModcclm 23238 ℂVecccvs 23299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-ip 16330 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-subg 17949 df-cmn 18555 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-subrg 19141 df-lmod 19228 df-lvec 19469 df-sra 19540 df-rgmod 19541 df-cnfld 20114 df-clm 23239 df-cvs 23300 |
This theorem is referenced by: (None) |
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